Deformation of finite-volume hyperbolic Coxeter polyhedra, limiting growth rates and Pisot numbers
Alexander Kolpakov1
Department of Mathematics, University of Fribourg, chemin du Musée 23, CH-1700 Fribourg, Switzerland
A connection between real poles of the growth functions for Coxeter groups acting on hyperbolic space of dimensions three and greater and algebraic integers is investigated. In particular, a certain geometric convergence of fundamental domains for cocompact hyperbolic Coxeter groups with finite-volume limiting polyhedron provides a relation between Salem numbers and Pisot numbers. Several examples conclude this work.
1. Introduction
Since the work of Steinberg [19], growth series for Coxeter groups are known to be series expansions of certain rational functions. By considering the growth function of a hyperbolic Coxeter group, being a discrete group generated by a finite setSof reflections in hyperplanes of hyperbolic space Hn, Cannon [2,3], Wagreich [23], Parry [14] and Floyd [8] in the beginning of the 1980s discovered a connection between the real poles of the corresponding growth function and algebraic integers such as Salem numbers and Pisot numbers forn = 2,3. In particular, there is a kind of geometric convergence for the fundamental domains of cocompact planar hyperbolic Coxeter groups giving a geometric interpretation of the convergence of Salem numbers to Pisot numbers, the behaviour discovered by Salem [16] much earlier in 1944. This paper provides a generalisation of the result by Floyd [8] to the three-dimensional case (cf.Theorem 5).
2. Preliminaries 2.1
LetGbe a finitely generated group with generating setSproviding the pair(G,S). In the following, we often writeGfor(G,S)assumingSis fixed. Define the word-norm · :G→NonGwith respect
E-mail addresses:kolpakov.alexander@gmail.com,aleksandr.kolpakov@unifr.ch.
1 Tel.: +41 026 300 9201; fax: +41 26 300 9744.
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toSbyg =min{n|gis a product ofnelements fromS∪S−1}. Denote byakthe number of elements inGof word-normk, and puta0=1 as usually done for the empty word.
Definition.The growth series of the groupG =(G,S)with respect to its generating setSisf(t):=
fS(t)=∞
k=0aktk.
The seriesf(t)has positive radius of convergence sinceak≤(2|S|)k. The reciprocal of the radius of convergence is calledthe growth rateτofG. IfGis a Coxeter group with its Coxeter generating set S(cf. [9]), thenf(t)is a rational function by [1,19], calledthe growth functionofG.
LetP ⊂Hn,n≥2, be a finite-volume hyperbolic polyhedron all of whose dihedral angles are submultiples ofπ. Such a polyhedron is called a Coxeter polyhedronand gives rise to a discrete subgroupG=G(P)of Isom(Hn)generated by the setSof reflections in the finitely many bounding hyperplanes ofP. We callG=G(P)a hyperbolic Coxeter group. In the following we will study the growth function ofG=(G,S)=G(P).
The most important tool in the study of the growth function of a Coxeter group is Steinberg’s formula.
Theorem 1 (Steinberg, [19]).Let G be a Coxeter group with generating set S. Then 1
fS(t−1) =
T∈F
(−1)|T|
fT(t) , (1)
whereF = {T ⊆S| the subgroup of G generated by T is finite}.
Consider the groupG=G(P) generated by the reflections in the bounding hyperplanes of a hyperbolic Coxeter polyhedronP. Denote byΩk(P),0 ≤ k ≤ n−1 the set of allk-dimensional faces ofP. Elements inΩk(P)fork=0, 1 andn−1 are called vertices, edges and facets (or faces, in casen=3) ofP, respectively.
Observe that all finite subgroups ofGare stabilisers of elements F ∈Ωk(P)for somek≥ 0. By the result of Milnor [12], the growth rate of a hyperbolic group is strictly greater than 1. Hence, the growth rate of the reflection groupG(P)isτ >1, ifPis compact, and the growth functionfS(t)has a pole in(0,1).
2.2
In the context of growth rates we shall look at particular classes of algebraic integers.
Definition.A Salem number is a real algebraic integerα >1 such thatα−1is an algebraic conjugate ofαand all the other algebraic conjugates lie on the unit circle of the complex plane. Its minimal polynomial overZis called a Salem polynomial.
Definition.A Pisot–Vijayaraghavan number, or a Pisot number for short, is a real algebraic integer β >1 such that all the algebraic conjugates ofβare in the open unit disc of the complex plane. The corresponding minimal polynomial overZis called a Pisot polynomial.
Recall that a polynomialP(t)is reciprocal ifP˜(t)=tdegPP(t−1)equalsP(t), and anti-reciprocal if
˜
P(t)equals−P(t). The polynomialP˜(t)itself is calledthe reciprocal polynomialofP(t). The following result is very useful in order to detect Pisot polynomials.
Lemma 1 (Floyd, [8]).Let P(t)be a monic polynomial with integer coefficients such that P(0) = 0, P(1) <0, and P(t)is not reciprocal. LetP˜(t)be the reciprocal polynomial for P(t). Suppose that for every sufficiently large integer m,tmP(tt−)−˜1P(t)is a product of cyclotomic polynomials and a Salem polynomial.
Then P(t)is a product of cyclotomic polynomials and a Pisot polynomial.
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Table 1
Coxeter exponents.
Vertex group Stab(v) Coxeter exponents
m1 m2 m3
Δ2,2,n,n≥2 1 1 n−1
Δ2,3,3 1 2 3
Δ2,3,4 1 3 5
Δ2,3,5 1 5 9
The convergence of Salem numbers to Pisot numbers was first discovered and analysed in [16]. A geometrical relation between these algebraic integers comes into view as follows. Growth functions of planar hyperbolic Coxeter groups were calculated explicitly in [8, Section 2]. The main result of [8]
states that the growth rateτof a cocompact hyperbolic Coxeter group – being a Salem number by [14]
– converges from below to the growth rate of a finite covolume hyperbolic Coxeter group under a certain deformation process performed on the corresponding fundamental domains. More precisely, one deforms the given compact Coxeter polygon by decreasing one of its anglesπ/m. This process results in pushing one of its vertices toward the ideal boundary∂H2in such a way that every polygon under this process provides a cocompact hyperbolic Coxeter group. Therefore, a sequence of Salem numbersαmgiven by the respective growth ratesτmarises. The limiting Coxeter polygon is of finite area having exactly one ideal vertex, and the growth rateτ∞ of the corresponding Coxeter group equals the limit ofβ=limm→∞αmand is a Pisot number.
2.3
In this work, we study analogous phenomena in the case of spatial hyperbolic Coxeter groups. The next result will play an essential role.
Theorem 2 (Parry, [14]).Let P ⊂ H3be a compact Coxeter polyhedron. The growth function f(t)of G(P)satisfies the identity
1
f(t−1) = t−1
t+1+
v∈Ω0(P)
gv(t), (2)
where
gv(t)= t(1−t) 2
(tm1−1)(tm2−1)(tm3−1)
(tm1+1−1)(tm2+1−1)(tm3+1−1) (3)
is a function associated with each vertexv∈Ω0(P); the integers m1,m2,m3are the Coxeter exponents2of the finite Coxeter groupStab(v)(seeTable1). Furthermore, the growth rateτof G(P)is a Salem number.
A rational functionf(t), that is not a polynomial, is called reciprocal iff(t−1) = f(t), and anti- reciprocal iff(t−1)= −f(t). In the case of growth functions for Coxeter groups acting cocompactly onHn, the following result holds.
Theorem 3 (Charney, Davis, [5]).Let G=(G,S)be a Coxeter group acting cocompactly onHn. Then, its growth function fS(t)is reciprocal if n is even, and anti-reciprocal if n is odd.
For further references on the subject, which treat several general and useful aspects for this work, we refer to [4,11].
The following example illustrates some facts mentioned above.
2 For the definition see, e.g. [9, Section 3.16, p. 75].
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Fig. 1. The dodecahedronDn⊂H3,n≥0, with all but one right dihedral angles. The specified angle equalsn+π2.
Example. LetDn⊂H3,n∈N, be a hyperbolic dodecahedron with all but one right dihedral angles.
The remaining angle along the thickened edge ofDn, as shown inFig. 1, equals nπ+2,n ≥ 0. The initial polyhedronD0is known as the Löbell polyhedronL(5)(see [20]). Asn → ∞, the sequence of polyhedra tends to a right-angled hyperbolic polyhedronD∞with precisely one vertex at infinity.
Let us compute the growth functions and growth rates ofG(Dn),n≥0, andG(D∞).
ByTheorem 2, the growth function ofG(Dn), with respect to the generating setSof reflections in the faces ofDn, equals
fn(t)= (1+t)3(1+t+ · · · +tn−1)
1−8t+8tn+1−tn+2 , (4)
and similarly
f∞(t)= (1+t)3
(1−t)(1−8t). (5)
Observe that the function(4)is anti-reciprocal, but the function(5)is not.
The computation of the growth ratesτn,n≥ 0, forG(Dn)and of the growth rateτ∞forG(D∞) gives
τ0≈7.87298< τ1≈7.98453<· · ·< τ∞=8.
Thus, the Salem numbersτn,n≥0, tend from below toτ∞, which is a Pisot number.
Consider a finite-volume polytopeP ⊂Hnand a compact faceF ∈Ωn−2(P)with dihedral angle αF. We always suppose thatP is not degenerated (i.e. not contained in a hyperplane). Suppose that there is a sequence of polytopesP(k)⊂Hnhaving the same combinatorial type and the same dihedral angles asP =P(1)apart fromαFwhose counterpartαF(k)tends to 0 ask ∞. Suppose that the limiting polytopeP∞exists and has the same number of facets asP. This means the facetF, which is topologically a codimension two ball, is contracted to a point, which is a vertex at infinityv∞ ∈∂Hn ofP∞. We call this processcontraction of the face F to an ideal vertex.
Remark.In the casen =2, an ideal vertex of a Coxeter polygonP0 ⊂H2comes from ‘‘contraction of a compact vertex’’ [8]. This means a vertexF ∈Ω0(P)of some hyperbolic Coxeter polygonPis pulled towards a point at infinity.
In the above deformation process, the existence of the polytopes P(k) in hyperbolic space is of fundamental importance. Let us consider the three-dimensional case. Since the angles of hyperbolic finite-volume Coxeter polyhedra are non-obtuse, the theorem by E.M. Andreev [22, p. 112, Theorem 2.8] is applicable in order to conclude about their existence and combinatorial structure.
In order to state Andreev’s result, recall that ak-circuit,k ≥ 3, is an ordered sequence of faces F1, . . . ,Fkof a given polyhedronP such that each face is adjacent only to the previous and the following ones, while the last one is adjacent only to the first one, and no three of them share a common vertex.
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Fig. 2. A ridge of typek1,k2,n,l1,l2.
Theorem 4 (Vinberg, [22]). LetP be a combinatorial polyhedron, not a simplex, such that three or four faces meet at every vertex. Enumerate all the faces of Pby1, . . . ,|Ω2(P)|. Let Fibe a face, Eij=Fi∩Fj
an edge, and Vijk= ∩s∈{i,j,k}Fsor Vijkl= ∩s∈{i,j,k,l}Fsa vertex of P. Letαij≥0be the weight of the edge Eij. The following conditions are necessary and sufficient for the polyhedronP to exist inH3having the dihedral anglesαij:
(m0) 0< αij≤π2.
(m1) If Vijkis a vertex ofP, thenαij+αjk+αki≥π, and if Vijklis a vertex ofP, thenαij+αjk+αkl+αli= 2π.
(m2) If Fi,Fj,Fkform a3-circuit, thenαij+αjk+αki< π.
(m3) If Fi,Fj,Fk, Flform a4-circuit, thenαij+αjk+αkl+αli<2π.
(m4) If Pis a triangular prism with bases F1and F2, thenα13+α14+α15+α23+α24+α25<3π. (m5) If among the faces Fi,Fj,Fk, the faces Fiand Fj,Fjand Fkare adjacent, Fiand Fkare not adjacent, but
concurrent in a vertexv∞, and all three Fi,Fj, Fkdo not meet atv∞, thenαij+αjk< π. 3. Coxeter groups acting on hyperbolic three-space
3.1. Deformation of finite volume Coxeter polyhedra
Let P ⊂ H3 be a Coxeter polyhedron of finite volume with at least five faces. Suppose that k1,k2,n,l1,l2≥2, are integers.
Definition. An edgee ∈ Ω1(P)is a ridge of typek1,k2,n,l1,l2ifeis bounded and has trivalent verticesv, wsuch that the dihedral angles at the incident edges are arranged counter-clockwise as follows: the dihedral angles along the edges incident tovarekπ1,kπ2 andπn, the dihedral angle along the edges incident towareπl1,πl2 andπn. In addition, the faces sharingeare at least quadrangles (see Fig. 2).
Note. Figs. 4–10 are drawn according to the following pattern: only significant combinatorial elements are highlighted (certain vertices, edges and faces), and the remaining ones are not specified and overall coloured grey. In each figure, the polyhedron is represented by its projection onto one of its supporting planes, and its dihedral angles of the formπ/mare labelled withm.
Proposition 1. LetP ⊂H3be a Coxeter polyhedron of finite volume with|Ω2(P)| ≥5. If Phas a ridge e∈Ω1(P)of type2,2,n,2,2,n≥2, then e can be contracted to a four-valent ideal vertex.
Proof. Denote byP(m)a polyhedron having the same combinatorial type and the same dihedral angles asP, except for the angleαm = mπ alonge. We show thatP(m)exists for allm ≥ n. Both verticesv, wofe ∈ Ω1(P(m))are points inH3, since the sum of dihedral angles at each of them equalsπ+mπform≥n≥2. Thus, conditionm1of Andreev’s theorem holds. Conditionm0is obviously satisfied, as well as conditionsm2−m4, sinceαm≤αn.
During the same deformation, the planes intersecting atebecome tangent to a pointv∞∈∂H3at infinity. The pointv∞is a four-valent ideal vertex with right angles along the incident edges. Denote the resulting polyhedron byP∞.
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Fig. 3. Two possible positions of the contracted edgee. The forbidden 3-circuit is dotted and forbidden prism bases are encircled by dotted lines.
Fig. 4. The first possible position of the contracted edgee. The forbidden 4-circuit is dotted. Face IV is at the back of the picture.
Fig. 5. The second possible position of the contracted edgee. The forbidden 3-circuit is dotted. Face III is at the back of the picture.
Since the contraction process deforms only one edge to a point, no new 3- or 4-circuits appear inP∞. Hence, for the existence ofP∞⊂H3 only conditionm5of Andreev’s theorem remains to be verified. Suppose that conditionm5is violated and distinguish the following two cases for the polyhedronPleading toP∞under contraction of the edgee.
1. Pis a triangular prism. There are two choices of the edgee∈Ω1(P), that undergoes contraction tov∈Ω∞(P∞), as shown inFig. 3on the left and on the right. SinceP∞is a Coxeter polyhedron, the violation ofm5implies that the dihedral angles along the edgese1ande2have to equalπ/2.
But then, either conditionm2orm4is violated, depending on the position of the edgee.
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Fig. 6. Two possible ridges resulting in a four-valent vertex under contraction.
Fig. 7. Pushing together and pulling apart the supporting planes of polyhedron’s faces results in an ‘‘edge contraction’’–‘‘edge insertion’’ process.
Fig. 8. Forbidden 3-circuit: the first case.
Fig. 9. Forbidden 3-circuit: the second case. The forbidden circuit going through the ideal vertex is dotted. Face III is at the back of the picture.
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Fig. 10. Forbidden 4-circuit. The forbidden circuit going through the ideal vertex is dotted. Face II is at the back of the picture.
2. Otherwise, the two possible positions of the edgeeinFigs. 4and5. The dihedral angles along the top and bottom edges are right, sincem5is violated after contraction.
2.1 Consider the polyhedronP inFig. 4on the right. SinceP is not a triangular prism, we may suppose (without loss of generality) that the faces I, II, III, IV in the picture are separated by at least one more face lying in the left grey region. But then, the faces I, II, III and IV ofPform a 4-circuit violating conditionm3of Andreev’s theorem.
2.2 Considerthe polyhedronPon the right inFig. 5. As before, we may suppose that the faces I, II, III form a 3-circuit. This circuit violates conditionm2of Andreev’s theorem forP.
Thus, the non-existence ofP∞ implies the non-existence ofP, and one arrives at a contradic- tion.
Note.Proposition 1describes the unique way of ridge contraction. Indeed, there is only one infinite family of distinct spherical Coxeter groups representing Stab(v), wherevis a vertex of the ridgee, and this one isΔ2,2,n,n ≥ 2. One may compare the above limiting process for hyperbolic Coxeter polyhedra with the limiting process for orientable hyperbolic 3-orbifolds from [6].
Proposition 2. LetP ⊂H3be a Coxeter polyhedron of finite volume with at least one four-valent ideal vertexv∞. Then there exists a sequence of finite-volume Coxeter polyhedraP(n) ⊂H3having the same combinatorial type and dihedral angles asP except for a ridge of type2,2,n,2,2, with n sufficiently large, giving rise to the vertexv∞under contraction.
Proof. Consider the four-valent ideal vertexv∞ofP and replacev∞by an edgeein one of the two ways as shown inFig. 6while keeping the remaining combinatorial elements ofPunchanged. Let the dihedral angle alongebe equal to πn, withn ∈Nsufficiently large. We denote this new polyhedron byP(n). The geometrical meaning of the ‘‘edge contraction’’–‘‘edge insertion’’ process is illustrated inFig. 7. We have to verify the existence ofP(n)inH3. Conditionsm0andm1of Andreev’s theorem are obviously satisfied forP(n). Conditionm5is also satisfied sincencan be taken large enough.
Suppose that one of the remaining conditions of Andreev’s theorem is violated. The inserted edge eofP(n)might appear in a new 3- or 4-circuit not present inPso that several cases are possible.
1. P(n)is a triangular prism. The polyhedronP(n)violating conditionm2of Andreev’s theorem is illustrated inFig. 3on the right. SinceP(n)is Coxeter, the 3-circuit depicted by the dashed line comprises the three edges in the middle, with dihedral anglesπn,π2 andπ2 along them. Contracting the edgeeback tov∞, we observe that conditionm5for the polyhedronPdoes not hold.
Since there are no 4-circuits, the only condition of Andreev’s theorem forP(n), which might be yet violated, ism4. This case is depicted inFig. 3on the left. A similar argument as above leads to a contradiction.
2. Otherwise, we consider the remaining unwanted cases, when either conditionm2or conditionm3
is violated.
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2.1 Case of a3-circuit. InFigs. 8and9, we illustrate two possibilities to obtain a 3-circuit inP(n) for allnsufficiently large, which violates conditionm2of Andreev’s theorem. The faces of the 3-circuit are indicated by I, II and III. InFig. 8, the edgeeis ‘‘parallel’’ to the circuit, meaning thatebelongs to precisely one of the faces I, II or III.
InFig. 9, the edgeeis ‘‘transversal’’ to the circuit, meaning thateis the intersection of precisely two of the faces I, II or III. Contracting eback tov∞ leads to an obstruction for the given polyhedronP to exist, as illustrated inFig. 8andFig. 9on the right. The polyhedronP in Fig. 8has two non-geometric faces, namely I and III, having in common precisely one edge and the vertexv∞disjoint from it. The polyhedronPinFig. 9violates conditionm5of Andreev’s theorem because of the triple, that consists of the faces I, II and III (inFig. 9on the right, face III is at the back of the picture).
2.2 Case of a4-circuit. First, observe that the sum of dihedral angles along the edges involved in a 4-circuit transversal to the edgeedoes not exceed 32π + πn, and therefore is less than 2πfor alln>2. This means conditionm3of Andreev’s theorem is always satisfied fornsufficiently large.
Finally, a 4-circuit parallel to the edgeeinP(n)is illustrated inFig. 10. The faces in this 4-circuit are indicated by I, II, III, IV. Suppose that the 4-circuit violates conditionm3. Contractingeback tov∞ (seeFig. 10on the right) leads to a violation ofm5forP because of the circuit, that consists of the faces I, II and III (inFig. 10on the right, the face II is at the back of the picture).
Note. The statements ofPropositions 1and2are essentially given in [22, p. 238] without proof. In the higher-dimensional case, no codimension two face contraction is possible. Indeed, the contraction process produces a finite-volume polytopeP∞⊂Hn,n≥4, whose volume is a limit point for the set of volumes ofP(k)⊂Hnask→ ∞. But, by the theorem of H.-C. Wang [22, Theorem 3.1], the set of volumes of Coxeter polytopes inHnis discrete ifn≥4.
4. Limiting growth rates of Coxeter groups acting onH3
The result of this section is inspired by Floyd’s work [8] on planar hyperbolic Coxeter groups. We consider a sequence of compact polyhedraP(n)⊂H3with a ridge of type2,2,n,2,2converging, asn→ ∞, to a polyhedronP∞with a single four-valent ideal vertex. According to [14], all the growth rates of the corresponding reflection groupsG(P(n))are Salem numbers. Our aim is to show that the limiting growth rate is a Pisot number.
The following definition will help us to make the technical proofs more transparent when studying the analytic behaviour of growth functions.
Definition. For a given Coxeter groupGwith generating setSand growth functionf(t)=fS(t), define F(t)=FS(t):= fS(t1−1).
Proposition 3. LetP∞ ⊂H3be a finite-volume Coxeter polyhedron with at least one four-valent ideal vertex obtained from a sequence of finite-volume Coxeter polyhedraP(n)by contraction of a ridge of type2,2,n,2,2as n→ ∞. Denote by fn(t)and f∞(t)the growth functions of G(P(n))and G(P∞), respectively. Then
1 fn(t)− 1
f∞(t) = tn 1−tn
1−t 1+t
2
.
Moreover, the growth rateτnof G(P(n))converges to the growth rateτ∞of G(P∞)from below.
Proof. We calculate the difference ofFn(t)andF∞(t)by means of Eq.(1). In fact, this difference is caused only by the stabilisers of the ridgee∈Ω1(P(n))and of its verticesvi ∈Ω0(P(n)),i=1,2.
Let[k] :=1+· · ·+tk−1. Here Stab(e)Dn, the dihedral group of order 2n, and Stab(vi)Δ2,2,n. The corresponding growth functions are given byfe(t)= [2][n]andfvi(t)= [2]2[n],i=1,2 (see [18]).
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Thus
Fn(t)−F∞(t)= f1
e(t)−f 1
v1(t)−f 1
v2(t) =tn1−1 t−1
t+1 2
. (6)
Next, perform the substitutiont →t−1on(6)and use the relation betweenFn(t),F∞(t)and their counterpartsfn(t) andf∞(t)according to the definition above. As a result, we obtain the desired formula, which yieldsf1
n(t)− f∞1(t) >0 fort∈(0,1).
Consider the growth ratesτnandτ∞ofG(P(n))andG(P∞). The least positive pole offn(t)is the least positive zero off1
n(t), andfn(0) =1. Similar statements hold forf∞(t). Hence, by the inequality above and by the definition of growth rate, we obtainτn−1> τ∞−1, orτn< τ∞, as claimed.
Finally, the convergenceτn→τ∞asn→ ∞follows from the convergencef1
n(t)− f∞1(t) →0 on (0,1), due to the first part of the proof.
Note.Given the assumptions ofProposition 3, the volume ofP(n)is less than that ofP∞by Schläfli’s volume differential formula [13]. Thus, growth rate and volume are both increasing under contraction of a ridge.
Consider two Coxeter polyhedraP1andP2inH3having the same combinatorial type and dihedral angles except for the respective ridgesH1= k1,k2,n1,l1,l2andH2= k1,k2,n2,l1,l2.
Definition.We say thatH1≺H2if and only ifn1<n2.
The following proposition extendsProposition 3to a more general context.
Proposition 4. Let P1 and P2 be two compact hyperbolic Coxeter polyhedra having the same combinatorial type and dihedral angles except for an edge of ridge type H1and H2, respectively. If H1≺H2, then the growth rate of G(P1)is less than the growth rate of G(P2).
Proof. Denote byf1(t)andf2(t)the growth functions ofG(P1)andG(P2), respectively. As before, we will show thatf1
1(t)−f21(t) ≥0 on(0,1).
Without loss of generality, we may suppose the ridgesHito be of typek1,k2,ni,l1,l2,i=1,2, up to a permutation of the sets{k1,k2},{l1,l2}and{{k1,k2},{l1,l2}}. By means ofTable 1showing all the finite triangle reflection groups, all admissible ridge pairs can be determined. We collected them inTable 2. The rest of the proof, starting with the computation off1
1(t)−f21(t)in accordance with Theorem 2, Eqs.(2)and(3), follows by analogy toProposition 3.
From now onP(n)always denotes a sequence of compact polyhedra inH3having a ridge of type 2,2,n,2,2, withnsufficiently large, that converges to a polyhedronP∞with a single four-valent ideal vertex. The corresponding growth functions for the groupsG(Pn)andG(P∞)are denoted by fn(t)andf∞(t). As above, we will work with the functionsFn(t)andF∞(t). ByTheorem 3, bothfn(t) andFn(t)are anti-reciprocal rational functions.
Consider the denominator of the right-hand side of Steinberg’s formula (cf.Theorem 1). According to [4, Section 5.2.2] and [11, Section 2.1], we introduce the following concept.
Definition.The lowest common multiple of the polynomialsfT(t),T ∈F, is called the virgin form of the numerator offS(t−1).
The next result describes the virgin form of the denominator ofF∞(t).
Proposition 5. LetP∞⊂H3be a polyhedron of finite volume with a single four-valent ideal vertex. Then the function F∞(t)related to the Coxeter group G(P∞)is given by
F∞(t)= t(t−1)P∞(t) Q∞(t) ,
where Q∞(t)is a product of cyclotomic polynomials,degQ∞(t)−degP∞(t) = 2, and P∞(0) = 0, P∞(1) <0.
